Parth Ajmera

Rate of heat transfer by conduction

\hat{Q} = -kA\frac{dT}{dy} = -kA\frac{T_2 - T_1}{y}

$\hat{Q}$ — rate of heat transfer (thermal energy crossing area $A$ per unit time), in watts ( $\text{W}$ )
$k$ — thermal conductivity of the wall, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )
$A$ — area perpendicular to heat transfer, in square metres ( $\text{m}^2$ )
$dT/dy$ — temperature gradient, in kelvin per metre ( $\text{K/m}$ )
$T_2, T_1$ — temperatures either side of the wall, in kelvin ( $\text{K}$ )
$y$ — distance component of thickness, in metres ( $\text{m}$ )

\hat{Q} = hA \Delta T

$\hat{Q}$ — rate of heat transfer (thermal energy crossing area $A$ per unit time), in watts ( $\text{W}$ )
$h$ — individual (film) heat transfer coefficient — how readily heat crosses between the bulk fluid and the wall by convection, in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$A$ — area, in square metres ( $\text{m}^2$ )
$\Delta T$ — temperature gradient, in kelvin ( $\text{K}$ )

Nu = \frac{hD}{k_{fb}}

$Nu$ — Nusselt number (ratio of convective to conductive heat transfer), dimensionless
$h$ — individual heat transfer coefficient, in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$D$ — diameter of the pipe or tank, or characteristic length of the pipe, in metres ( $\text{m}$ )
$k_{fb}$ — thermal conductivity of the bulk fluid, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )

Pr = \frac{C_p \mu_b}{k_{fb}}

$Pr$ — Prandtl number (ratio of molecular momentum to thermal diffusivity), dimensionless
$C_p$ — heat capacity, in joules per kilogram-kelvin ( $\text{J}/(\text{kg} \cdot \text{K})$ )
$\mu_b$ — viscosity of the bulk fluid, in pascal-seconds ( $\text{Pa} \cdot \text{s}$ )
$k_{fb}$ — thermal conductivity of the bulk fluid, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )

R_{cond} = \frac{x}{k}

$R_{cond}$ — conductive thermal resistance per unit area, in square metre-kelvin per watt ( $\text{m}^2 \cdot \text{K}/\text{W}$ )
$x$ — thickness of the layer, in metres ( $\text{m}$ )
$k$ — thermal conductivity of the layer's material, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )

R_{total} = \sum_i \frac{x_i}{k_i}

$R_{total}$ — total conductive resistance of the composite wall per unit area, in square metre-kelvin per watt ( $\text{m}^2 \cdot \text{K}/\text{W}$ )
$x_i, k_i$ — thickness and thermal conductivity of layer $i$ , in metres ( $\text{m}$ ) and watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ ) respectively

\hat{Q} = UA \Delta T_{LMTD}

$\hat{Q}$ — rate of heat transfer (thermal energy crossing area $A$ per unit time), in watts ( $\text{W}$ )
$U$ — overall heat transfer coefficient (lumps both film coefficients and the wall's own conduction resistance into one number, avoiding the need to measure $\Delta T$ at each internal interface), in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$A$ — area, in square metres ( $\text{m}^2$ )
$\Delta T_{LMTD}$ — logarithmic mean temperature difference, in kelvin ( $\text{K}$ )

\frac{1}{U} = \frac{1}{h_1} + \frac{x}{k} + \frac{1}{h_2} + R_f

$U$ — overall heat transfer coefficient (lumps both film coefficients and the wall's own conduction resistance into one number), in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$h_1, h_2$ — individual film heat transfer coefficients on either side of the wall, in watts per square metre-kelvin ( $\text{W}/(\text{m}^2 \cdot \text{K})$ )
$x$ — thickness of the wall, in metres ( $\text{m}$ )
$k$ — thermal conductivity of the wall, in watts per metre-kelvin ( $\text{W}/(\text{m} \cdot \text{K})$ )
$R_f$ — fouling resistance at a film surface (e.g. microbial biofilm or scale buildup), in square metre-kelvin per watt ( $\text{m}^2 \cdot \text{K}/\text{W}$ )

\Delta T_{LMTD} = \frac{\Delta T_2 - \Delta T_1}{\ln(\Delta T_2 / \Delta T_1)}

$\Delta T_{LMTD}$ — logarithmic mean temperature difference (single equivalent driving-force temperature for the whole exchanger, replacing the $\Delta T$ that actually varies along its length), in kelvin ( $\text{K}$ )
$\Delta T_2, \Delta T_1$ — temperature differences at the two ends of the heat exchanger, in kelvin ( $\text{K}$ )

Q_{metabolic} = \Delta H_{O_2} \times n_{O_2}

$Q_{metabolic}$ — metabolic heat released by aerobic respiration, in kilojoules ( $\text{kJ}$ )
$\Delta H_{O_2}$ — heat released per mole of oxygen consumed (**Cooney's correlation**), approximately $460\,\text{kJ/mol}$ ( $\approx 14\,\text{kJ/g}$ $O_2$ , since $460/32 \approx 14.4$ )
$n_{O_2}$ — moles of oxygen consumed by the culture ( $\text{mol}$ )

\frac{A}{V} \propto \frac{D^2}{D^3} = \frac{1}{D}

$A$ — cooling surface area available (jacket or coil), in square metres ( $\text{m}^2$ )
$V$ — reactor working volume, in cubic metres ( $\text{m}^3$ )
$D$ — characteristic vessel diameter, in metres ( $\text{m}$ )